AMD-Numbers, Compactness, Strict Singularity and the Essential Spectrum of Operators

Abstract

For an operator 𝑇 acting from an infinite-dimensional Hilbert space 𝐻 to a normed space 𝑌 we define the upper AMD-number and the lower AMD-number as the upper and the lower limit of the net (δ(𝑇|𝐸))𝐸∈𝐹𝐷(𝐻), with respect to the family 𝐹𝐷(𝐻) of all finite-dimensional subspaces of 𝐻. When these numbers are equal, the operator is called AMD-regular. It is shown that if an operator 𝑇 is compact, then and, conversely, this property implies the compactness of 𝑇 provided 𝑌 is of cotype 2, but without this requirement may not imply this. Moreover, it is shown that an operator 𝑇 has the property if and only if it is superstrictly singular. As a consequence, it is established that any superstrictly singular operator from a Hilbert space to a cotype 2 Banach space is compact. For an operator 𝑇, acting between Hilbert spaces, it is shown that and are respectively the maximal and the minimal elements of the essential spectrum of , and that 𝑇 is AMD-regular if and only if the essential spectrum of |𝑇| consists of a single point.

MD-Numbers and Asymptotic MD-Numbers of Operators

“…Notice, finally, that the numbers δ 1 (T ), δ 1 (T ) and δ 1 (T ) can also be defined when Y is a general Banach space. The properties of the corresponding quantities in this general setting are investigated in [2].…”